The partial Rigidity Conjecture for consecutive inverse coefficients

Let $m,n\in\mathbb N_+$ and let $a(X)\in{\mathbb C}[X]$ be a polynomial of degree at most $m+1$ such that $a(X)\equiv X\pmod{X^2}$. Let $a^{-1}(X)$ be its formal inverse for composition. Partial Rigidity Conjecture. If the coefficients of $X^{n+1},\ldots,X^{n+m}$ in $a^{-1}(X)$ vanish, then $a(X)=X$. This is the parameterized form introduced after the general Rigidity Conjecture. The paper proves some low-dimensional cases, but does not resolve the assertion in general.